I've had to deal with rambunctious students. One in particular, has turned into quite a talented pianist. She's is about 12 now, though I started as her second teacher when she was about seven. She definitely didn't sit still, often laying down on the bench, acting silly, ect. But she's gradually evened out over the years, so maybe it is just a maturity thing. I mean, as many times as she had been lectured about acting correctly, she didn't really change until *she wanted too*.
Your student sounds like a challenging, though immature, student. And challenging students can be fun. I think the idea of having her watch youtube videos is a really great idea. Also, maybe setting aside time for her to be dramatic? Maybe have her compose a song that tells a story and that she also has to act while playing?
And how do you mean she is agitated? Is she full of energy or does she just get upset when you make a comment? I have one or two students that don't really like comments from me. Both are smart and know that they have done something wrong and would rather work out the problem during the week and just play their pieces at their lesson. This is again, the maturity thing. They are just not ready for constructive criticism and would rather just focus on practicing. When they become ready, they will ask me to help them. (It's happened.) Or, they will learn by themselves to fix the problem and I will give them praise for that.
Also, maybe instead of telling her the problem, ask her if there is anything she did wrong or needs to work on.
And, if she really can't sit still because she has lots of energy, then try 15 minute lessons. Once or twice a week.